View Single Post
fred4321
fred4321 is offline
#1
Oct30-11, 03:19 AM
P: 4
Hi,

I am having trouble understanding how this works.

I am giving the following:
y[k+2] - y[k+1] + 0.24y[k] = f[k+2] - 2f[k+1];

y[-2] = 1, y[-1] = 2;

f[k] = 0 for k < 0;
f[k] = k for k >= 0;

I would like to have a program compute the next values in the sequence, so, I need y[-2] = 1 to become y[1] = 1 and y[-1] = 2 to become y[2] = 1 (so that the array indexing works, e.g., I can access a negative location of an array).

I let k' = k + 1 so that I'd get:
y[k'+3] - y[k'+2] + 0.24y[k'+1] = f[k'+3] - 2f[k'+1];
Then I made:
f[k] = 0 for k < 0;
f[k] = k for k >= 0;
become
f[k'] = 0 for k' < 3;
f[k'] = k for k' >= 3;
and
y[-2] = 1, y[-1] = 2;
become
y[1] = 1, y[2] = 2;

So now I have:
y[k'+3] - y[k'+2] + 0.24y[k'+1] = f[k'+3] - 2f[k'+1];
f[k'] = 0 for k' < 3;
f[k'] = k for k' >= 3;
y[1] = 1, y[2] = 2;

And now when I let k = 0, k[3] gives me the value that k[0] gave me in the old equation, which is exactly what I want.

My issue is, I don't understand how, mathematically, this works. For example, I don't understand how I went from:
f[k] = 0 for k < 0;
f[k] = k for k >= 0;
to
f[k'] = 0 for k' < 3;
f[k'] = k for k' >= 3;

if k' = k + 1.

It seems as though I've shifted the equation (y[k+2] - y[k+1] + 0.24y[k] = f[k+2] - 2f[k+1];) over by 1 unit in the positive x direction; however, I've shifted the initial values (y[-2] = 1, y[-1] = 2;) and the f's restrictions (k < 0; and k >= 0) over by 3 units.


What I'm thinking is:
The original question should be:
f[k'] = 0 for k < k[0];
f[k'] = k for k' >= k[0];
y[k[0]-2] = 1, y[k[0]-1] = 2;

What would be the correct, more formal approach to achieving what I want. Also, should the original question be as I've written above?

I'm pretty sure that the equation, as it's give, only produces the 'correct' answer, when k[0] = -2.

Thank you for your time, I realise that this question is rather lengthy.
Phys.Org News Partner Mathematics news on Phys.org
Math modeling handbook now available
Hyperbolic homogeneous polynomials, oh my!
Researchers help Boston Marathon organizers plan for 2014 race